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Identifying the vertex of a parabola is an important first step in graphing. For a quadratic equation in standard form, such as \( y = ax^2 + bx + c \), the vertex provides a crucial point of reference. The vertex

• is a turning point of the parabola.
• determines whether the parabola opens upwards or downwards depending on the sign of \( a \).

To find the vertex \((h, k)\), use the formula for \( h \):\[ h = \frac{-b}{2a} \]Plugging in the values from the equation \( y = -2x^2 - 2x - 6 \), we calculate:\[ h = \frac{-(-2)}{2(-2)} = -\frac{1}{2} \]
Then, substitute back to determine \( k \):\[ k = y(-\frac{1}{2}) = -2\left(-\frac{1}{2}\right)^2 - 2\left(-\frac{1}{2}\right) - 6 = -\frac{11}{2} \] Thus, the vertex is \((-\frac{1}{2}, -\frac{11}{2})\).This tells you the graph turns at this point and provides symmetry for plotting.

The y-intercept is where the graph of the parabola crosses the y-axis. This is a significant point because it is straightforward to determine and plot. To find the y-intercept, simply replace \( x \) with 0 in the equation. Thus for \( y = -2x^2 - 2x - 6 \), the calculation is:
\[ y = -2(0)^2 - 2(0) - 6 = -6 \]The y-intercept is therefore \((0, -6)\). This point tells us the value of \( y \) when \( x = 0 \), and it's directly plotted on the y-axis. The y-intercept gives one point to help draw the parabola and understand its intersection with the y-axis.

Additional coordinate points supplement the vertex and y-intercept when sketching a parabola. They give more structure to the graph you're about to draw. Choose any values for \( x \) to calculate corresponding \( y \) values.
1. When \( x = -1 \): \[ y = -2(-1)^2 - 2(-1) - 6 = -6 \] Point: \((-1, -6)\)
2. When \( x = 1 \): \[ y = -2(1)^2 - 2(1) - 6 = -10 \] Point: \((1, -10)\)These points provide additional frames for the graph and help make the shape more accurate. Placing them with the vertex and y-intercept helps create a fuller picture of the parabola, revealing its downward opening due to the negative \( a \) value in the equation.

Once you have plotted the vertex, y-intercept, and additional points, it's time to draw the parabola. But how do you ensure accuracy? Using a graphing calculator is a fantastic way to verify your work.
Here's how it works:

• Input the quadratic equation: \( y = -2x^2 - 2x - 6 \) into the calculator.
• The calculator generates the curve and displays it on the screen.
• Compare this with your hand-drawn version to see if they align.

Graphing calculators allow you to quickly check your work, ensuring fidelity to the mathematical model.They offer a powerful visual confirmation of your effort in sketching the parabola.